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Nicolas Chopin

Publications and source records attributed to Nicolas Chopin.

At least 55 records · Page 3Linked to original sources
arXiv2015

On Particle Methods for Parameter Estimation in State-Space Models

Nonlinear non-Gaussian state-space models are ubiquitous in statistics, econometrics, information engineering and signal processing. Particle methods, also known as Sequential Monte Carlo (SMC) methods, provide reliable numerical approximations to the associated state inference problems. However, in most applications, the state-space model of interest also depends on unknown static parameters that need to be estimated from the data. In this context, standard particle methods fail and it is necessary to rely on more sophisticated algorithms. The aim of this paper is to present a comprehensive review of particle methods that have been proposed to perform static parameter estimation in state-space models. We discuss the advantages and limitations of these methods and illustrate their performance on simple models.

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arXiv2015

On particle Gibbs sampling

The particle Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm to sample from the full posterior distribution of a state-space model. It does so by executing Gibbs sampling steps on an extended target distribution defined on the space of the auxiliary variables generated by an interacting particle system. This paper makes the following contributions to the theoretical study of this algorithm. Firstly, we present a coupling construction between two particle Gibbs updates from different starting points and we show that the coupling probability may be made arbitrarily close to one by increasing the number of particles. We obtain as a direct corollary that the particle Gibbs kernel is uniformly ergodic. Secondly, we show how the inclusion of an additional Gibbs sampling step that reselects the ancestors of the particle Gibbs' extended target distribution, which is a popular approach in practice to improve mixing, does indeed yield a theoretically more efficient algorithm as measured by the asymptotic variance. Thirdly, we extend particle Gibbs to work with lower variance resampling schemes. A detailed numerical study is provided to demonstrate the efficiency of particle Gibbs and the proposed variants.

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arXiv2015

Leave Pima Indians alone: binary regression as a benchmark for Bayesian computation

Abstract. Whenever a new approach to perform Bayesian computation is introduced, a common practice is to showcase this approach on a binary regression model and datasets of moderate size. This paper discusses to which extent this practice is sound. It also reviews the current state of the art of Bayesian computation, using binary regression as a running example. Both sampling-based algorithms (importance sampling, MCMC and SMC) and fast approximations (Laplace and EP) are covered. Extensive numerical results are provided, some of which might go against conventional wisdom regarding the effectiveness of certain algorithms. Implications for other problems (variable selection) and other models are also discussed.

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arXiv2015

Convergence of Sequential Quasi-Monte Carlo Smoothing Algorithms

Gerber and Chopin (2015) recently introduced Sequential quasi-Monte Carlo (SQMC) algorithms as an efficient way to perform filtering in state-space models. The basic idea is to replace random variables with low-discrepancy point sets, so as to obtain faster convergence than with standard particle filtering. Gerber and Chopin (2015) describe briefly several ways to extend SQMC to smoothing, but do not provide supporting theory for this extension. We discuss more thoroughly how smoothing may be performed within SQMC, and derive convergence results for the so-obtained smoothing algorithms. We consider in particular SQMC equivalents of forward smoothing and forward filtering backward sampling, which are the most well-known smoothing techniques. As a preliminary step, we provide a generalization of the classical result of Hlawka and Mück (1972) on the transformation of QMC point sets into low discrepancy point sets with respect to non uniform distributions. As a corollary of the latter, we note that we can slightly weaken the assumptions to prove the consistency of SQMC.

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arXiv2015

On the properties of variational approximations of Gibbs posteriors

The PAC-Bayesian approach is a powerful set of techniques to derive non- asymptotic risk bounds for random estimators. The corresponding optimal distribution of estimators, usually called the Gibbs posterior, is unfortunately intractable. One may sample from it using Markov chain Monte Carlo, but this is often too slow for big datasets. We consider instead variational approximations of the Gibbs posterior, which are fast to compute. We undertake a general study of the properties of such approximations. Our main finding is that such a variational approximation has often the same rate of convergence as the original PAC-Bayesian procedure it approximates. We specialise our results to several learning tasks (classification, ranking, matrix completion),discuss how to implement a variational approximation in each case, and illustrate the good properties of said approximation on real datasets.

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arXiv2015

Towards automatic calibration of the number of state particles within the SMC$^2$ algorithm

SMC$^2$ is an efficient algorithm for sequential estimation and state inference of state-space models. It generates $N_θ$ parameter particles $θ^{m}$, and, for each $θ^{m}$, it runs a particle filter of size $N_{x}$ (i.e. at each time step, $N_{x}$ particles are generated in the state space $\mathcal{X}$). We discuss how to automatically calibrate $N_{x}$ in the course of the algorithm. Our approach relies on conditional Sequential Monte Carlo updates, monitoring the state of the pseudo random number generator and on an estimator of the variance of the unbiased estimate of the likelihood that is produced by the particle filters, which is obtained using nonparametric regression techniques. We observe that our approach is both less CPU intensive and with smaller Monte Carlo errors than the initial version of SMC$^2$.

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arXiv2015

Application of Sequential Quasi-Monte Carlo to Autonomous Positioning

Sequential Monte Carlo algorithms (also known as particle filters) are popular methods to approximate filtering (and related) distributions of state-space models. However, they converge at the slow $1/\sqrt{N}$ rate, which may be an issue in real-time data-intensive scenarios. We give a brief outline of SQMC (Sequential Quasi-Monte Carlo), a variant of SMC based on low-discrepancy point sets proposed by Gerber and Chopin (2015), which converges at a faster rate, and we illustrate the greater performance of SQMC on autonomous positioning problems.

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arXiv2014

Sequential Quasi-Monte Carlo

We derive and study SQMC (Sequential Quasi-Monte Carlo), a class of algorithms obtained by introducing QMC point sets in particle filtering. SQMC is related to, and may be seen as an extension of, the array-RQMC algorithm of L'Ecuyer et al. (2006). The complexity of SQMC is $O(N \log N)$, where $N$ is the number of simulations at each iteration, and its error rate is smaller than the Monte Carlo rate $O_P(N^{-1/2})$. The only requirement to implement SQMC is the ability to write the simulation of particle $x_t^n$ given $x_{t-1}^n$ as a deterministic function of $x_{t-1}^n$ and a fixed number of uniform variates. We show that SQMC is amenable to the same extensions as standard SMC, such as forward smoothing, backward smoothing, unbiased likelihood evaluation, and so on. In particular, SQMC may replace SMC within a PMCMC (particle Markov chain Monte Carlo) algorithm. We establish several convergence results. We provide numerical evidence that SQMC may significantly outperform SMC in practical scenarios.

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arXiv2014

The Poisson transform for unnormalised statistical models

Contrary to standard statistical models, unnormalised statistical models only specify the likelihood function up to a constant. While such models are natural and popular, the lack of normalisation makes inference much more difficult. Here we show that inferring the parameters of a unnormalised model on a space $Ω$ can be mapped onto an equivalent problem of estimating the intensity of a Poisson point process on $Ω$. The unnormalised statistical model now specifies an intensity function that does not need to be normalised. Effectively, the normalisation constant may now be inferred as just another parameter, at no loss of information. The result can be extended to cover non-IID models, which includes for example unnormalised models for sequences of graphs (dynamical graphs), or for sequences of binary vectors. As a consequence, we prove that unnormalised parameteric inference in non-IID models can be turned into a semi-parametric estimation problem. Moreover, we show that the noise-contrastive divergence of Gutmann & Hyvärinen (2012) can be understood as an approximation of the Poisson transform, and extended to non-IID settings. We use our results to fit spatial Markov chain models of eye movements, where the Poisson transform allows us to turn a highly non-standard model into vanilla semi-parametric logistic regression.

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arXiv2014

Bayesian matrix completion: prior specification

Low-rank matrix estimation from incomplete measurements recently received increased attention due to the emergence of several challenging applications, such as recommender systems; see in particular the famous Netflix challenge. While the behaviour of algorithms based on nuclear norm minimization is now well understood, an as yet unexplored avenue of research is the behaviour of Bayesian algorithms in this context. In this paper, we briefly review the priors used in the Bayesian literature for matrix completion. A standard approach is to assign an inverse gamma prior to the singular values of a certain singular value decomposition of the matrix of interest; this prior is conjugate. However, we show that two other types of priors (again for the singular values) may be conjugate for this model: a gamma prior, and a discrete prior. Conjugacy is very convenient, as it makes it possible to implement either Gibbs sampling or Variational Bayes. Interestingly enough, the maximum a posteriori for these different priors is related to the nuclear norm minimization problems. We also compare all these priors on simulated datasets, and on the classical MovieLens and Netflix datasets.

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arXiv2014

PAC-Bayesian AUC classification and scoring

We develop a scoring and classification procedure based on the PAC-Bayesian approach and the AUC (Area Under Curve) criterion. We focus initially on the class of linear score functions. We derive PAC-Bayesian non-asymptotic bounds for two types of prior for the score parameters: a Gaussian prior, and a spike-and-slab prior; the latter makes it possible to perform feature selection. One important advantage of our approach is that it is amenable to powerful Bayesian computational tools. We derive in particular a Sequential Monte Carlo algorithm, as an efficient method which may be used as a gold standard, and an Expectation-Propagation algorithm, as a much faster but approximate method. We also extend our method to a class of non-linear score functions, essentially leading to a nonparametric procedure, by considering a Gaussian process prior.

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arXiv2012

Bayesian learning of noisy Markov decision processes

We consider the inverse reinforcement learning problem, that is, the problem of learning from, and then predicting or mimicking a controller based on state/action data. We propose a statistical model for such data, derived from the structure of a Markov decision process. Adopting a Bayesian approach to inference, we show how latent variables of the model can be estimated, and how predictions about actions can be made, in a unified framework. A new Markov chain Monte Carlo (MCMC) sampler is devised for simulation from the posterior distribution. This step includes a parameter expansion step, which is shown to be essential for good convergence properties of the MCMC sampler. As an illustration, the method is applied to learning a human controller.

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arXiv2012

Computational aspects of Bayesian spectral density estimation

Gaussian time-series models are often specified through their spectral density. Such models present several computational challenges, in particular because of the non-sparse nature of the covariance matrix. We derive a fast approximation of the likelihood for such models. We propose to sample from the approximate posterior (that is, the prior times the approximate likelihood), and then to recover the exact posterior through importance sampling. We show that the variance of the importance sampling weights vanishes as the sample size goes to infinity. We explain why the approximate posterior may typically multi-modal, and we derive a Sequential Monte Carlo sampler based on an annealing sequence in order to sample from that target distribution. Performance of the overall approach is evaluated on simulated and real datasets. In addition, for one real world dataset, we provide some numerical evidence that a Bayesian approach to semi-parametric estimation of spectral density may provide more reasonable results than its Frequentist counter-parts.

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arXiv2012

Bayesian nonparametric estimation of the spectral density of a long or intermediate memory Gaussian process

A stationary Gaussian process is said to be long-range dependent (resp., anti-persistent) if its spectral density $f(λ)$ can be written as $f(λ)=|λ|^{-2d}g(|λ|)$, where $0<d<1/2$ (resp., $-1/2<d<0$), and $g$ is continuous and positive. We propose a novel Bayesian nonparametric approach for the estimation of the spectral density of such processes. We prove posterior consistency for both $d$ and $g$, under appropriate conditions on the prior distribution. We establish the rate of convergence for a general class of priors and apply our results to the family of fractionally exponential priors. Our approach is based on the true likelihood and does not resort to Whittle's approximation.

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arXiv2012

Expectation-Propagation for Likelihood-Free Inference

Many models of interest in the natural and social sciences have no closed-form likelihood function, which means that they cannot be treated using the usual techniques of statistical inference. In the case where such models can be efficiently simulated, Bayesian inference is still possible thanks to the Approximate Bayesian Computation (ABC) algorithm. Although many refinements have been suggested, ABC inference is still far from routine. ABC is often excruciatingly slow due to very low acceptance rates. In addition, ABC requires introducing a vector of "summary statistics", the choice of which is relatively arbitrary, and often require some trial and error, making the whole process quite laborious for the user. We introduce in this work the EP-ABC algorithm, which is an adaptation to the likelihood-free context of the variational approximation algorithm known as Expectation Propagation (Minka, 2001). The main advantage of EP-ABC is that it is faster by a few orders of magnitude than standard algorithms, while producing an overall approximation error which is typically negligible. A second advantage of EP-ABC is that it replaces the usual global ABC constraint on the vector of summary statistics computed on the whole dataset, by n local constraints of the form that apply separately to each data-point. As a consequence, it is often possible to do away with summary statistics entirely. In that case, EP-ABC approximates directly the evidence (marginal likelihood) of the model. Comparisons are performed in three real-world applications which are typical of likelihood-free inference, including one application in neuroscience which is novel, and possibly too challenging for standard ABC techniques.

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arXiv2012

In praise of the referee

There has been a lively debate in many fields, including statistics and related applied fields such as psychology and biomedical research, on possible reforms of the scholarly publishing system. Currently, referees contribute so much to improve scientific papers, both directly through constructive criticism and indirectly through the threat of rejection. We discuss ways in which new approaches to journal publication could continue to make use of the valuable efforts of peer reviewers.

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arXiv2012

Fast simulation of truncated Gaussian distributions

We consider the problem of simulating a Gaussian vector X, conditional on the fact that each component of X belongs to a finite interval [a_i,b_i], or a semi-finite interval [a_i,+infty). In the one-dimensional case, we design a table-based algorithm that is computationally faster than alternative algorithms. In the two-dimensional case, we design an accept-reject algorithm. According to our calculations and our numerical studies, the acceptance rate of this algorithm is bounded from below by 0.5 for semi-finite truncation intervals, and by 0.47 for finite intervals. Extension to 3 or more dimensions is discussed.

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